A Unified Explanation of the Kadowaki--Woods Ratio in Strongly Correlated Metals

LETTERS PUBLISHED ONLINE: 19 APRIL 2009 | DOI: 10.1038/NPHYS1249

A uniﬁed explanation of the Kadowaki–Woods ratio in strongly correlated metals A. C. Jacko1, J. O. Fjærestad2 and B. J. Powell1*

Discoveries of ratios whose values are constant within broad 104 K-NCS classes of materials have led to many deep physical insights. K-Br The Kadowaki–Woods ratio (KWR; refs 1, 2) compares the 103 UBe13 temperature dependence of a metal’s resistivity to that of its 2 10 CeCu heat capacity, thereby probing the relationship between the 6 1 CeCu2Si2 electron–electron scattering rate and the renormalization of 10 ⊥ Sr2RuO4 LiV2O4 the electron mass. However, the KWR takes very different ) 0 10 Na0.7CoO2 UPt3 3,4 ¬2 values in different materials . Here we introduce a ratio, CeB6 ¬1 β-I closely related to the KWR, that includes the effects of 10 3 β-IBr UAl2 cm K 2 Tl2Ba2CuO6 carrier density and spatial dimensionality and takes the same μΩ ¬2 Rb3C60

( 10 (predicted) value in organic charge-transfer salts, transition- A ⎢⎢ Sr2RuO4 metal oxides, heavy fermions and transition metals—despite 10¬3 La1.7Sr0.3CuO4 the numerator and denominator varying by ten orders of ¬4 Heavy fermions 10 Pd Organics magnitude. Hence, in these materials, the same emergent Fe Transition metals physics is responsible for the mass enhancement and the 10¬5 Re Ni Pt Oxides quadratic temperature dependence of the resistivity, and no a 10¬6 Os TM exotic explanations of their KWRs are required. aHF In a Fermi liquid the electronic contribution to the heat capacity 100 101 102 103 104 105 106 γ has a linear temperature dependence, that is, Cel(T) = T. Another γ 2 (mJ2 mol¬2 K¬4) prediction of Fermi-liquid theory5 is that, at low temperatures, 2 the resistivity varies as ρ(T) = ρ0 + AT . This is observed Figure 1 | The standard Kadowaki–Woods plot. It can be seen that the experimentally when electron–electron scattering, which gives rise data for the transition metals and heavy fermions (other than UBe13) to the quadratic term, dominates over electron–phonon scattering. fall onto two separate lines. However, a wide range of other strongly 1 2 In a number of transition metals A/γ ≈ aTM = 0.4 µ cm correlated metals do not fall on either line or between the two lines. 2 2 −2 2 2 2 −2 mol K J (Fig. 1), even though γ varies by an order of magnitude aTM = 0.4 µ cm mol K J is the value of the KWR observed in the 2 1 2 2 −2 across the materials studied. Later, it was found that in many transition metals and aHF = 10 µ cm mol K J is the value seen 2 2 2 −2 2 heavy-fermion compounds A/γ ≈ aHF = 10 µ cm mol K J in the heavy fermions . In labelling the data points we use the following 2 (Fig. 1), despite the large mass renormalization, which causes γ abbreviations: κ-Br is κ-(BEDT-TTF)2Cu[N(CN)2]Br; κ-NCS is to vary by more than two orders of magnitude in these materials. κ-(BEDT-TTF)2Cu(NCS)2; β-I3 is β-(BEDT-TTF)2I3; and β-IBr2 is 2 Because of this remarkable behaviour A/γ has become known as β-(BEDT-TTF)2IBr2. For Sr2RuO4 we show data for A measured with the the Kadowaki–Woods ratio. However, it has long been known2,6 current both perpendicular and parallel to the basal plane; these data that the heavy-fermion material UBe13 has an anomalously large points are distinguished by the symbols ⊥ and k respectively. Further KWR. More recently, studies of other strongly correlated metals, details of the data are reported in Supplementary Information. such as the transition-metal oxides3,7 and the organic charge- transfer salts4,8, have found surprisingly large KWRs (Fig. 1). It There have been a number of studies of the KWR based on is therefore clear that the KWR is not the same in all metals; in specific microscopic Hamiltonians (see, for example, refs 9–13). fact, it varies by more than seven orders of magnitude across the However, if the KWR has something general to tell us about materials shown in Fig. 1. strongly correlated metals, then we would also like to under- Several important questions need to be answered about the stand which features of the ratio transcend specific microscopic KWR. (1) Why is the ratio approximately constant within the models. Nevertheless, two important points emerge from these transition metals and within the heavy fermions (even though microscopic treatments of the KWR: (1) if the momentum dep- many-body effects cause large variations in their effective masses)? endence of the self-energy can be ignored then the many-body (2) Why is the KWR larger for the heavy fermions than it is renormalization effects on A and γ 2 cancel, and (2) material- for the transition metals? (3) Why are such large and varied specific parameters are required to reproduce the experimentally KWRs observed in layered metals such as the organic charge- observed values of the KWR. Below we investigate the KWR transfer salts and transition-metal oxides? The main aim of using a phenomenological Fermi-liquid theory; this work builds this paper is to resolve question (3). We shall also make on previous studies of related models3,6. Indeed, our calculation some comments on the first two questions, which have been is closely related to that in ref. 6. The main results reported extensively studied previously. here are the identification of a ratio (equation (5)) relating A

1Centre for Organic Photonics and Electronics, School of Physical Sciences, University of Queensland, Brisbane, Queensland 4072, Australia, 2School of Physical Sciences, University of Queensland, Brisbane, Queensland 4072, Australia. *e-mail: [email protected].

γ 2 and , which we predict takes a single value in a broad class 104 Transition metals K-NCS of strongly correlated metals, and the demonstration that this Heavy fermions 103 Organics K-Br ratio does indeed describe the data for a wide variety of strongly UBe13 correlated metals (Fig. 2). Oxides 102 It has been argued that the KWR is larger in the heavy CeCu6 fermions than the transition metals because the former are 1 10 CeCu2Si2 Sr RuO ⊥ more strongly correlated (in the sense that the self-energy is LiV2O4 2 4 6 ) UPt more strongly frequency dependent) than the latter . Several ¬2 100 Na0.7CoO2 3 β-I scenarios have been proposed to account for the large KWRs 3 CeB6 cm K 10¬1 UAl2 β observed in UBe13, transition-metal oxides and organic charge- -IBr2 6 μΩ transfer salts, including impurity scattering , proximity to a ( Tl2Ba2CuO6 A ¬2 7 10 Sr RuO ⎢⎢ quantum critical point and the suggestion that electron–phonon 2 6 Rb3C60 scattering in reduced dimensions might give rise to a quadratic 10¬3 La Sr CuO temperature dependence of the resistivity14. It has been previously 1.7 0.3 4 3 observed that using volumetric (rather than molar) units for 10¬4 Pd γ reduces the variation in the KWRs of the transition-metal Ni oxides. However, even in these units, the organic charge-transfer 10¬5 Pt Fe Re salts have KWRs orders of magnitude larger than those of other Os strongly correlated metals. We shall argue that the different 10¬132 10¬130 10¬128 10¬126 10¬124 10¬122 γ 2 4 9 ¬6 ¬4 KWRs observed across this wide range of materials result from /fdx (n) (kg m s K ) the simple fact that the KWR contains a number of material- specific quantities. As a consequence, when we replace the KWR Figure 2 | Comparison of the ratio deﬁned in equation (5) with with a ratio that accounts for these material-specific effects experimental data. It can be seen that, in all of the materials studied, (equation (5)) the data for all of these materials do indeed lie on the data are in excellent agreement with our prediction (line). The a single line (Fig. 2). abbreviations in the data-point labels are the same as in Fig. 1. Further Many properties of strongly correlated Fermi liquids can be details of the data are given in Supplementary Information. understood in terms of a momentum-independent self-energy15,16. Therefore, following ref. 6, we assume that the imaginary part of the (DOS) at the Fermi energy. In the low-temperature, pure limit we self-energy, Σ 00(ω,T), at energy ω, is given by find (see the Methods section) that

2 ω2 ( )2 16nkB 00 ~ + πkBT A = (3) Σ (ω,T) = − −s (1) 2h 2 i 2ω∗2 ∗2 π~e v0x D0 2τ0 ω where h···i denotes an average over the Fermi surface. Note that 2 2 ∗2 for |ω +(πkBT) | < ω and neither the DOS nor the Fermi velocity are renormalized in this expression. Indeed, all of the many-body effects are encapsulated ∗ 00 / ∗ by ω , which determines the magnitude of the frequency-dependent Σ (ω,T) = −[~/2τ +s]F([ω2 +(πk T)2]1 2/ω ) 0 B term in Σ 00(ω,T); see equation (1). The Kramers–Kronig relation for the retarded self-energy19,20 2 2 ∗2 for |ω + (πkBT) | > ω , where 2s/~ is the scattering rate can be used to show (see the Methods section) that, in the due to electron–electron scattering in the absence of quantum pure limit, τ −1 many-body effects, 0 is the impurity scattering rate, F is a monotonically decreasing function with boundary conditions ∂Σ 0 4s ξ F(1) = 1 and F(∞) = 0 and ω∗ is determined by the strength of γ = γ 1− = γ 1+ u 0 ∂ω 0 πω∗ the many-body correlations. (See the Methods section for further discussion of the self-energy.) 17 γ = 2 2 / The diagonal part of the conductivity tensor may be written as where 0 π kBD0 3 is the linear coefficient of the specific heat for a gas of non-interacting fermions, Σ 0 is the real part ξ ≈ Z dk Z dω −∂f (ω) of the self-energy and 1 is a pure number defined in the σ = 2 2 2 ,ω γ xx (T) ~e v0x As (k ) (2) Methods section. Thus we see that the renormalization of is also π 3 π ∂ω ∗ (2 ) 2 controlled by ω . For a strongly correlated metal the effective mass, ∗ ∗ m m0, the bare (band) mass of the electron, hence su ω and = , , = −1∂ε /∂ γ ' 2ξ / ω∗ where k (kx ky kz ) is the momentum, v0x ~ 0(k) kx is the (8nkB ) (9 ). The corrections to this approximation are given unrenormalized velocity in the x direction, f (ω) is the Fermi–Dirac in the Methods section. ∗ −1 distribution, As(k, ω) = −2 Im{[ω − ε0(k) + µ − Σ(ω, T)] } Combining the above results we see that the KWR is is the spectral density, ε0(k) is the non-interacting dispersion relation and µ∗ is the chemical potential. Note that equation (2) A 81 1 = (4) γ 2 2 2 ξ 2 2 2 does not contain vertex corrections; the absence of vertex 4π~kBe nD0hv0x i corrections to the conductivity is closely related to the momentum independence of the self-energy15. Further, the presence of First, we note that in this ratio the dependence of the individual Umklapp processes, which enable electron–electron scattering to factors on ω∗ has vanished. Hence the KWR is not renormalized. On contribute to the resistivity in the pure limit18, is implicit in the other hand, although the first factor contains only fundamental the above formula. constants, the second factor is clearly material dependent as it In a strongly correlated metal, s may be approximated by its depends on the electron density, the DOS and the Fermi velocity 6,16 value in the unitary scattering limit , su = 2n/3πD0, where n is of the non-interacting system. An important corollary to this result the conduction-electron density and D0 is the bare density of states is that band-structure calculations should give accurate predictions

NATURE PHYSICS | VOL 5 | JUNE 2009 | www.nature.com/naturephysics 423 © 2009 Macmillan Publishers Limited. All rights reserved. LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS1249 of the KWR by equation (4), as none of the properties on the The k-space integral here is (half) the renormalized DOS, D∗, at the renormalized ∗ = / right-hand side are renormalized. Fermi energy. As D D0 Z, where D0 is the DOS of the bare system at its The wide range of KWRs found in layered materials suggests Fermi energy, we get ξ 2 2h 2 i that nD0 v0x varies significantly in these materials. For example, Z −∂f (ω)/∂ω 2 σ = 2h 2 i ω for highly anisotropic materials hv i may vary by more than an xx (T) ~e v0x D0 d (7) 0x −2Σ 00(ω,T) order of magnitude, depending on which direction the resistivity is measured in. This effect needs to be taken into account if we Using −∂f (ω)/∂ω → δ(ω) as T → 0 it follows from equation (1) that the wish to understand what the KWR has to tell us about strongly ρ = 2τ h 2 i −1 zero-temperature resistivity is given by 0 (e 0 v0x D0) . With the temperature 2 correlated metals. Further, equation (4) suggests that the reason dependence of the resistivity given by ρ(T)=ρ0 +AT , equation (7) yields that the transition metals and the heavy fermions have ‘constant’ ξ 2 2h 2 i " #−1  KWRs is that nD0 v0x is roughly constant across each class of 1 Z −∂f (ω)/∂ω ~ 2 2 2 2 ρ ρ ω 6= ξ h i AT = − 0 =  d −  materials and that aHF aTM because nD0 v0x is different in the ~e2hv2 iD −2Σ 00(ω,T) τ two different classes of materials. Therefore, we propose that a more 0x 0 0 fundamental ratio is We now consider the pure limit, τ0 → ∞. At sufficiently low temperatures the contribution to the integral from the region ω > ω∗ is small, so it is a good Afdx (n) 81 00 ω, ω = (5) approximation to use equation (1) for Σ ( T) for all . The resulting integral γ 2 2 2 can then be evaluated analytically, 4π~kBe

≡ 2h 2 iξ 2 ∞ ∗2 ∞ ∗ 2 where fdx (n) nD0 v0x may be written in terms of the Z (−∂f (ω)/∂ω) ω Z (−∂f (ω)/∂ω) (ω /k T) dω ≈ dω = B 00 2 2 dimensionality, d, of the system, the electron density and, in layered −∞ −2Σ (ω,T) 2s −∞ ω +(πkBT) 24s systems, the interlayer spacing or the interlayer hopping integral. For simplicity we assume that the reasonably isotropic materials Equation (3) follows on taking s = su. (the heavy fermions, the transition metals and Rb3C60) are isotropic Fermi liquids and the layered materials (that is, the Real part of the self-energy. To calculate γ we need to know the real part of 19,20 transition-metal oxides and organic charge-transfer salts based the self-energy. To evaluate this we apply the Kramers–Kronig relation for the retarded self-energy, on BEDT-TTF) have warped cylindrical Fermi surfaces; fdx (n) is derived for these band structures in the Methods section. 1 Z ∞ Σ 00(ω0,T) 0 ω, = 0 ∞, + ω0 Σ ( T) Σ ( T) P 0 d This enables us to test explicitly, in Fig. 2, the prediction of π −∞ ω −ω equation (5) against previously published experimental data for a variety of strongly correlated metals. It is clear that the where P indicates the principal part of the integral. ∗ new ratio is in good agreement with the data for all of the For T =0 and taking the pure limit we find that for |ω|ω materials investigated. We therefore see that the observations of  1 Z ∞ Σ 00(ω0) 2s 1 1−y constant KWRs for the heavy fermions and for the transition ω0 = − + 2 P 0 d y y ln metals are due not only to the profound but also to the π −∞ ω −ω π 2 1+y prosaic. The renormalization of γ 2 cancels with that of A owing  Z ∞ 0 ∞ 2n+1 to the Kramers–Kronig relation for the self-energy; but the 0 F(y ) X y + dy unrenormalized properties are remarkably consistent within each 0 0  1 y n=0 y class of materials. Further, the large KWRs in transition-metal oxides, the organics and UBe13 are simply a consequence of the where y = ω/ω∗, y0 = ω0/ω∗, and we have used the fact that F(y) is an even small values of fdx (n) in these materials. Therefore, the absolute function. Therefore, in the limit |y| 1 value of the KWR does not reveal anything about electronic 4s ξ ω ω3 correlations unless the material-specific effects, described by fdx (n), Σ 0(ω,0) = Σ 0(∞,0)− u +O are first accounted for. π ω∗ ω∗3

It will be interesting to identify and understand strongly ∞ ∞ where 1 < 2ξ ≡ 1 + R y−2F(y) dy ≤ 1 + R y−2 dy = 2 as F(y) ≤ 1 for y ≥ 1. correlated metals that are not described by equation (5). Our 1 1 Provided that F(y) decreases sufficiently slowly as y → ∞, we expect ξ ≈ 1. In calculation already gives some clues as to when this might general, changes to the exact form of the self-energy will simply lead to small happen: for example, when the self-energy is strongly momentum changes in ξ provided that the boundary conditions for Σ remain the same. Note dependent or when there are significant vertex corrections to the that Σ 0(∞,0) is just the shift in the zero-temperature chemical potential due to conductivity. Another outstanding challenge is to understand the many-body interactions. ω = ω∗ KWR in compensated semimetals21–23. The form of the self-energy given in equation (1) has a kink at . Kinks are found in the self-energies of local Fermi-liquid theories, such as dynamical mean-field theory24. However, the location, and even the existence, of this kink is Methods not important for our results. To calculate A and γ we must integrate over all ω Resistivity. To calculate A we approximate the spectral density by (see above); therefore, any sharp features in Σ 00(ω,T) will be washed out. Band structure. A, γ and n are relatively straightforward to determine 2πZδ(ω −Zξ (k)) h 2 i 2 ,ω ≈ 0 experimentally. It is harder to directly measure D0 and v0x . Therefore, we consider As (k ) 00 (6) ε 2 2/ −Σ (ω,T) two model band structures. (1) For an isotropic Fermi√ liquid 0(k) = ~ k 2m0, = / 2 2 h 2 i = 2 2/ 2 = 3 2 ξ = D0 m0kF ~ π and v0x ~ kF 3m0, where kF 3π n. Hence, for 1, p3 7 4 6 f3x (n) = 3n /π ~ . (2) To study layered materials we use the simple model where ξ0(k) = ε0(k)−µ0 with µ0 the chemical potential of the bare system (that 2 2 dispersion ε0(k) = ~ k /2m0 − 2t⊥0 cosck⊥, where k = (kab,k⊥), kab is the is, in the absence of both electron–electron interactions and impurity scattering) ab in-plane wavevector, k⊥ is the wavenumber perpendicular to the plane, c is the −1 0 and Z is the quasi-particle weight defined as Z = 1 − (∂/∂ω)Σ (ω,0)|ω=0. 00 interlayer spacing and t⊥0 is the bare interlayer hopping integral. If A is taken Equation (6) will give the right behaviour in the limit Σ → 0 (ref. 17). Inserting h 2 i = 2 2/ 2 from measurements of the resistivity parallel to the plane v0k ~ kF 2m0. equation (6) into equation (2) and noting that at low temperatures −∂f (ω)/∂ω 2 2 2 / 2 However, for A measured perpendicular to the plane hv ⊥i = 2c t⊥ ~ . In will be sharply peaked at ω = 0, we replace ω by zero in δ(ω −Zξ (k)). The delta 0 0 0 either case, for the warped√ cylindrical Fermi surface we are now considering function can then be taken outside the ω integration, giving 2 2 2 D0 = m0/πc~ and kF = 2πcn. So, for ξ = 1, we find that f2k(n) = n /πc~ and = 2 2 / 2 6 f2⊥(n) 2nm0t⊥0 π ~ . Z dk Z 1 −∂f (ω) σ (T) ≈ Z~e2hv2 i δ(Zξ (k)) dω This formalism can straightforwardly be generalized to include other factors xx 0x 3 0 00 (2π) −Σ (ω,T) ∂ω known to affect the KWR by extending the definition of fdx (n). For example, it

424 NATURE PHYSICS | VOL 5 | JUNE 2009 | www.nature.com/naturephysics © 2009 Macmillan Publishers Limited. All rights reserved. NATURE PHYSICS DOI: 10.1038/NPHYS1249 LETTERS has been shown13 that in the N -fold orbitally degenerate periodic Anderson model 14. Strack, Ch. et al. Resistivity studies under hydrostatic pressure on a A/γ 2 ∝ [N (N −1)]−1, in good agreement with experiments on heavy fermions low-resistance variant of the quasi-two-dimensional organic superconductor 25,26 with orbital degeneracy . This result is specific to this particular model, and the κ-(BEDT-TTF)2Cu[N(CN)2]Br: Search for intrinsic scattering contributions. 25,26 ξ 2 2h 2 i systems so far studied all have rather similar values of nD0 v0x . However, it is Phys. Rev. B 72, 054511 (2005). clear, from a comparison of the calculation of ref. 13 with ours, that if orbitally de- 15. Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J. Dynamical mean-field generate systems with different electron densities or reduced dimensionalities were theory of strongly correlated fermion systems and the limit of infinite fabricated fdx (n) would need to be included to understand the relationship between dimensions. Rev. Mod. Phys. 68, 13–125 (1996). A and γ . It has also been shown3 that the number of sheets of the Fermi surface 16. Hewson, A. C. The Kondo Problem To Heavy Fermions (Cambridge Univ. also affects the KWR. It is straightforward to generalize the above calculations of Press, 1997). fdx (n) to Fermi surfaces with any number of sheets. Finally, we note that if we relax 17. Mahan, G. D. Many-Particle Physics 2nd edn (Plenum Press, 1990). ∗ /γ 2 = / 2 2 − / ∗ 2 the condition m m0 then we find that Afdx (n) (81 4π~kBe )(1 m0 m ) , 18. Maebashi, H. & Fukuyama, H. Electrical conductivity of interacting fermions ∗ that is, the ratio vanishes as m → m0. 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